The quasi-Newton update proposed in Algorithm REF  differs from bfgs and other methods in the literature to ensure the following three properties are satisfied: i) \(P_b\)  is symmetric, and ii) positive definite with strictly positive eigenvalues, iii) Assumption REF  holds when \(\Vert \theta _b - \hat{\theta }_n\Vert \)  is small. Conditions i-ii) are required to ensure optimization is stable and apply Lemma REF . Condition iii) is needed for valid inference. Other methods do not satisfy i), ii), and iii) simultaneously. Amongst the most widely used methods, bfgs enforces i) and ii) plus the so-called secant equation: \(P_b y_{k} = s_k\)  so that \(P_b H_n(\theta _k)s_{k} = s_k\)  always holds but is only guaranteed in this direction so that iii) need not hold for non-quadratic \(Q_n\) .See [1]} for counter-examples. The BFGS matrix \(P_k\)  can have negative eigenvalues when \(Q_n\)  is non-convex. This is problematic for convergence as illustrated in Section REF . The sr1 update, used with trust-region algorithms, satisfies i) and iii) but not ii) which makes it unstable in estimation using (REF ).See [2]}, [3]} for derivations of iii) with SR1. Both methods update \(P_k\)  from \(P_{k-1}\)  using a rank-one update based on the secant equation above. Here \(P_b\)  is fully updated using OLS, this is related to the multi-secant update in [4]} which can satisfy iii) but typically not i-ii), and requires \(L \le d_{\theta }\) .
